Abstract: Quasar spectra carry the imprint of foreground intergalactic medium (IGM) through absorption systems. In particular, absorption caused by neutral hydrogen gas, the "Ly$\alpha$ forest," is a key spectroscopic tracer for cosmological analyses used to measure cosmic expansion and test physics beyond the standard model. Despite their importance, current methods for measuring Ly$\alpha$ absorption require making strong assumptions on the shape of the intrinsic quasar continuum that bias Ly$\alpha$ analyses. We present SpenderQ, a ML-based approach for reconstructing intrinsic quasar spectra and measuring the Ly$\alpha$ forest from observations. SpenderQ uses the Spender spectrum autoencoder to learn a compact and redshift-invariant latent encoding of quasar spectra, combined with an iterative procedure to identify and mask absorption regions in the spectra. We apply SpenderQ to 28,000 quasar spectra in the Early Data Release of the Dark Energy Spectroscopic Instrument and illustrate that it can reconstruct the intrinsic spectra of quasars, including the detailed features of broad emission lines (e.g., Ly$\beta$, Ly$\alpha$, SiIV, CIV, and CIII). Our method ignores dense absorption features in the Ly$\alpha$ forest and recovers the quasar continuum. SpenderQ provides a new data-driven approach for unbiased Ly$\alpha$ forest measurements in cosmological, quasar, and IGM studies.